Consider a system of N "weighted" coins, with tails likely to be showing only 40% of the time. (a) Derive a general expression for P(n), the probability of n coins showing heads, as a function of n, N, and p, the probability of tails showing. (b) For N = 10 "fair" coins, what value of p should you use? What are the probabilities that each possible number of heads is showing? Check your answers as compared to the less general expression used in class, i.e., "old" vs. "new". Check also that the sum of the probabilities is correct. What is the most likely number of heads showing? (c) For N = 10 "weighted" coins, as described above, what are the probabilities that each possible number of heads is showing? Check that the sum of the probabilities is correct. What is the most likely number of heads showing? (d) Make a single chart showing the histograms of both "fair" and "weighted" probabilities, as functions of the number of heads showing, as computed in parts (b) and (c). For part (a), your expression is worked out on paper, scanned, turned into a PDF, and combined with your other PDFs. The expression should be completely general, i.e., it should depend on n, N, and p. You should start with the expression for P(n), weighted, taken from the lecture notes. Make sure to define n, N, p, and q. (And don't be too surprised if the derivation is very short!) For part (b), start with the Excel file from class, and adapt it for N = 10 coins, removing all unused charts and worksheets. Make sure the total probability check is working before doing anything else. Remove all IF statements and unused formulas. There should be a column of probabilities already there, but it is not general enough to use for "weighted" coins. Therefore, add a column next to it. In the space above this column, insert the probabilities of heads and tails, then use your expression from part (a) to write an expression for the probability of 0 heads showing (the first macrostate). This expression should be general enough to copy both down and to the right, with $ symbols in all the right places